A generalized a-Wright convexity and related functional equation

Abstract

Let I be an interval and M,N : I x I I some means with the strict internality property. Suppose that <p : J —• R is a non-constant and continuous solution of the functional equation <p(M(x, y)) + ip(N(x, y)) = <p(x) + <p{y). Then tp is one-to-one; moreover for every lower semicontinuous function / : / - » R satisfying the inequality f(M(x,y)) + f(N(x,y)) < /(*) + f(y), the function / o ip~l is convex on <p(I). This is a generalization of an earlier result of Zs. Pales. An application to the a-Wright convex function is given.